Love the swimsuit! Very 1940's bombshell/pinup cute. Actually when I cover up a suit I usually pull shorts on - denim cutoffs, and/or a very light/airy white cotton button up shirt with rolled sleeves and long tails.
I think this suit would look great with dark denim shorts or rolled up skinny clam diggers. You could try white or black or navy for example, or alternately a flippy or pencil skirt in a neutral color or dark denim. If you are going for the 1940's look slightly high-waisted bottoms will match that.
I would think about using a cropped denim jacket or very light open short cardigan as a coverup, or just a very lightweight cotton white or chambray shirt if you are still in the sun but want your arms covered.
Here are some examples of the general idea I'm thinking of:
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